Using Theorem (REF ), \( \mathcal {N}_{\delta , \theta }\)  is closed for each \(\delta , \theta \in \Delta \) , and then by applying Proposition REF , it is clear that for each \(\delta , \theta \in \Delta \)  the space \(\mathcal {N}_{\delta ,\theta }\)  is an MI space. Also, the sampling operator \(T\)  is one-to-one on \(\bigcup _{\delta ,\theta \in \Delta }\mathcal {N}_{\delta ,\theta }\)  if and only if \(T\)  is one-to-one on \(\mathcal {N}_{\delta , \theta }\) 
for each \(\delta ,\theta \in \Delta \)  [1]}. Finally, applying Proposition REF  will give us the desired result.
